Constants Gap Weight Projection Phi Dftenergy Total Nonneg

Energy is never negative

In signal analysis, the total energy of a pattern is the sum of the energies carried by each of its frequency components. Each component's energy is the square of its amplitude, and a square is never negative. So the total, being a sum of non-negative pieces, is itself never negative. The declaration phiDFTEnergyTotal_nonneg is a machine-checked proof of exactly this fact for a specific pattern called the φ-pattern, which is built from the golden ratio.

The proof works by unpacking the definition of the total energy, which is written as a sum over the eight frequency modes of a discrete Fourier transform. The machine-checked library of formal theorems shows that each of the eight amplitudes is non-negative, and therefore their sum is non-negative. This is not a numerical approximation; it is a formal, verified statement that holds for all inputs, not just for one particular calculation.

This lemma is a small but necessary piece of a larger construction. The framework uses this total energy as a denominator when projecting a raw weight onto the eight-tick basis. A denominator that could be zero or negative would make the projection ill-defined or meaningless. This proof guarantees that the denominator is never negative, so the projection is always a sensible real number.

In Recognition Science, the framework models a discrete record of events, called a ledger, and the cost of recognizing a pattern. The φ-pattern is a specific sequence of values on this ledger. The total energy here is the sum of the squared magnitudes of the pattern's frequency components, a standard measure in signal processing. The lemma does not claim that this total energy is positive, only that it is greater than or equal to zero. It also does not claim that the projection weight itself is positive, only that it is non-negative, which is a separate lemma.

The practical consequence is that the projection weight, which converts a dimensionless fraction into a per-cell weight using a factor of 64, is always a valid real number. This is a hygiene result: it removes one possible source of undefined behavior in the framework's calculations. It is a foundational step, not a final result, and it does not by itself establish any physical law or numerical value.

lemma phiDFTEnergyTotal_nonneg : 0 ≤ phiDFTEnergyTotal := by
  unfold phiDFTEnergyTotal
  apply Finset.sum_nonneg
  intro k _
  exact phiDFTAmplitude_nonneg k

/-- Total DFT energy of the φ-pattern (Parseval denominator). -/
noncomputable def phiDFTEnergyTotal : ℝ :=
  Finset.univ.sum fun k : Fin 8 => phiDFTAmplitude k

/-- **Projection weight** of the φ-pattern onto the 8-tick basis:

`projectionScale * (rawWeightedNeutralEnergy / totalEnergy)`.

This makes the normalization and measure choice explicit. -/
noncomputable def w8_projected : ℝ :=
  projectionScale * (w8_dft_candidate / phiDFTEnergyTotal)

What this page does not claim

This lemma does not claim that the total energy is positive, only that it is non-negative. This lemma does not claim that the projection weight w8_projected is positive; that is a separate lemma. This lemma does not establish any physical law or numerical value for any constant.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Constants/GapWeight/Projection.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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